Optical fiber mode couplers

ABSTRACT

Described are optical devices and related methods wherein a multiple mode input in a Higher Order Mode (an HOM) optical fiber is converted by a complex mode transformer to produce a fundamental mode output in an optical medium with an E-field that is substantially different than that exiting the HOM optical fiber. The medium is preferably a Large Mode Area (LMA) optical fiber, or free space. The mode transformer may be a series of refractive index perturbations created either by photo-induced gratings or by gratings formed by physical deformations of the optical fiber.

RELATED APPLICATION

This application is related to application Ser. No. 12/157,214, filedJun. 9, 2008, which application is incorporated herein by reference.

FIELD OF THE INVENTION

The invention relates to optical fiber mode controlling devices.

BACKGROUND OF THE INVENTION

Optical fiber and optical waveguide mode converters are well known andcome in a variety of forms. They operate typically by transforming aninput mode, usually a fundamental mode, into a higher order mode, orvice versa. An especially attractive mode converter device comprises along period grating (LPG) formed in an optical fiber. See for example,U.S. Pat. No. 6,768,835, and T. Erdogan, “Fiber grating spectra,” J.Lightwave Technology vol. 15, p. 1277 (1997).

These mode converters operate with a single mode input, and typically asingle mode output. Propagating light in more than one mode at a time,and controllably changing the mode of more than one mode at a time,would be an attractive goal, but to date not widely achieved.

The function of effective and controlled mode conversion is useful indevices that process optical signals in higher order modes (HOMs). See,for example, U.S. Pat. No. 6,768,835, issued to Siddharth Ramachandranon Jul. 24, 2004, and incorporated by reference herein. A problem insome devices of this kind is that the radial dependence of the E-fieldsof the HOMs in the HOM optical fiber is complicated and not very usefulfor applications that require specific E-field distributions. Forinstance, it may be desirable to deliver extremely high power laserenergy through air-core photonic bandgap optical fibers (ACPBGs) orlarge mode area optical fibers (LMAs) as near Gaussian modes. However,such modes may have a very different E-field profile than the E-fieldprofile within the HOM optical fiber. In another example, tight focusingof the output of the HOM optical fiber may be required. The desired beamshape for focusing is typically a Gaussian free-space mode. The beamshape exiting the HOM optical fiber is typically far from Gaussian.dA ₁ /d _(z) =iσA ₁ +iκA ₁  Equation (5)

SUMMARY OF THE INVENTION

I have designed optical devices and related methods wherein a multiplemode input in a Higher Order Mode (an HOM) optical fiber is converted bya complex mode transformer to produce a fundamental mode output in anoptical medium with an E-field that is substantially different than thatexiting the HOM optical fiber. The medium is preferably a Large ModeArea (LMA) optical fiber, or free space. The mode transformer may be aseries of refractive index perturbations created either by photo-inducedgratings or by gratings formed by physical deformations of the opticalfiber.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is an illustration of a combination HOM and mode transformercoupled to a large mode area (LMA) optical fiber;

FIG. 2 is an illustration similar to that of FIG. 1 of a combination HOMand mode transformer coupled to free space;

FIG. 3 shows the combination of FIG. 2 with a focusing element;

FIG. 4 is a schematic representation of a mode transformer usingserially arranged multiple LPG mode transformers;

FIG. 5 is an illustration similar to that of FIG. 4 showing a modetransformer with superimposed multiple LPG transformers;

FIG. 6 shows a mode transformer diagram and a schematic form of a modetransformer based on index perturbations created by deforming an opticalfiber;

FIG. 7 is a schematic representation of multiple mode transformers basedon index perturbations created by deforming an optical fiber;

FIG. 8 shows a mode transformer diagram and a schematic form of a modetransformer which is a modified version of the multiple modetransformers of FIG. 7; and

FIG. 9 shows far field distribution profiles of multiple output modesproduced from a single input mode; and

FIG. 10 shows near field distribution profiles of multiple output modesproduced from a single input mode and is incorporated by reference fromU.S. application Ser. No. 12/157,214.

DETAILED DESCRIPTION

FIG. 1 is an illustration of a combination HOM optical fiber and modetransformer, coupled to a large mode area (LMA) optical fiber. The HOMoptical fiber is sometimes referred to as a few mode optical fiber. TheHOM optical fiber is shown at 11, with schematic HOM waveform 14illustrating the light propagating in the core 12 of the optical fiber.A fundamental mode (LP01) is shown in phantom for comparison.

The light signal 14, or another waveform (not shown) processed toproduce waveform 14, may be processed in the core of the HOM opticalfiber 11. The processing may be amplification, filtering, engineeredchromatic dispersion, etc. An example of processing HOMs in few modefibers is described in U.S. Pat. No. 6,768,835, referenced above.

Referring again to FIG. 1, the light signal 14 is then converted by modeconverter 15. The output 19 of the mode converter is coupled to an LMAoptical fiber 17. The core of the LMA fiber is shown at 18 and, as seen,is much larger than the core 12 of the HOM optical fiber. The modeconverter 15 desirably converts the waveform 14 to a waveform 19 thatmore closely matches the E-field of the LMA optical fiber.

Similar considerations apply to the case where the receiving opticalfiber is an ACPBG in place of the LMA optical fiber. The LMA embodimentis preferred but substituting an ACPBG is a useful alternative.

FIG. 2 shows a similar arrangement for propagating a near Gaussianwaveform 19, 19 a, in free space. This beam shape is desirable fordelivering the light energy to a remote location, for example,delivering high power laser output to a specific location or spotthrough an optical fiber pigtail. It is also desirable for focusing thebeam 19 in free space.

FIG. 3 shows the combination of HOM optical fiber 11 and mode converter15, as in FIG. 1, attached to a focusing element 31, in thisillustration a GRIN lens. The focusing element may be a useful accessoryfor shaping the output of the mode converter 15 to a desired shape 39.

Design of the mode converter shown at 15 in FIGS. 1-3 is described asfollows.

The simplest case of coupling between several copropagating modes iscoupling between two modes. Conversion between two modes can beperformed with a long period grating (LPG), which periodically changesthe effective refractive index of the fiber according to the followingequation:

$\begin{matrix}{{n_{eff}(z)} = {n_{0} + {\Delta\; n\;{\cos( {{\frac{2\;\pi}{\Lambda}z} + \theta} )}}}} & (1)\end{matrix}$where Λ is the period of the LPG. Assume that the LPG starts at z=0 andends at z=L (see FIG. 1). Consider modes 1 and 2 having the propagationconstants β₁ and β₂, respectively. For determinacy, assume that β₂>β₁.In the absence of LPG, at z<0, modes 1 and 2 have the form:E ₁(x,y,z)=C ₁₀exp(iβ ₁ z+iφ ₁)e ₁(x,y)E ₂(x,y,z)=C ₂₀exp(iβ ₂ z+iφ ₂)e ₂(x,y)′  (2)Here z is the coordinate along the fiber, x, y are the transversecoordinates, e_(j)(x,y) are the real-valued transverse modedistribution, and C_(j0) and φ_(j) are constants, which determine theamplitudes and the phases of modes, respectively. When these modes enterthe section of the fiber containing the LPG, the coordinate dependencecan be written in the form:

$\begin{matrix}{{{E_{1}( {x,y,z} )} = {{A_{1}(z)}\exp\{ {{{{\mathbb{i}}\lbrack {\beta_{1} - \delta + {\frac{1}{2}( {\sigma_{11} + \sigma_{22}} )}} \rbrack}z} + {\frac{\mathbb{i}}{2}\theta}} \}{e_{1}( {x,y} )}}}{{{E_{2}( {x,y,z} )} = {{A_{2}(z)}\exp\{ {{{{\mathbb{i}}\lbrack {\beta_{2} + \delta + {\frac{1}{2}( {\sigma_{11} + \sigma_{22}} )}} \rbrack}z} - {\frac{\mathbb{i}}{2}\theta}} \}{e_{2}( {x,y} )}}},{where}}} & (3) \\{{\delta = {{\frac{1}{2}( {\beta_{1} - \beta_{2}} )} + \frac{\pi}{\Lambda}}},} & (4)\end{matrix}$σ_(jj) are the “dc” coupling coefficients [see e.g. T. Erdogan, “Fibergrating spectra,” J. Lightwave Technology vol. 15, p. 1277 (1997)], andA_(j)(z) are the functions, which are determined by the coupling waveequations:

$\begin{matrix}{{\frac{\mathbb{d}A_{1}}{\mathbb{d}z} = {{{\mathbb{i}\sigma}\; A_{1}} + {{\mathbb{i}}\;\kappa\; A_{2}}}}{\frac{\mathbb{d}A_{2}}{\mathbb{d}z} = {{{\mathbb{i}}\;\kappa\; A_{2}} - {{\mathbb{i}}\;\sigma\; A_{2}}}}} & (5)\end{matrix}$Here σ is the general “dc” self-coupling coefficient and κ is the “ac”cross-coupling coefficient. Comparing Eq. (2) and Eq. (3), the initialconditions for A_(j)(z) are:

$\begin{matrix}{{{A_{1}(0)} = {C_{10}{\exp\lbrack {{\mathbb{i}}( {\varphi_{1} - \frac{\theta}{2}} )} \rbrack}}}{{A_{2}(0)} = {C_{20}{\exp\lbrack {{\mathbb{i}}( {\varphi_{2} + \frac{\theta}{2}} )} \rbrack}}}} & (6)\end{matrix}$Solution of Eq. (5) is:

$\begin{matrix}{{A_{1}(z)} = {{{( {{\cos( {\mu\; z} )} + {{\mathbb{i}}\;\frac{\sigma}{\mu}{\sin( {\mu\; z} )}}} ){A_{1}(0)}} + {{\mathbb{i}}\;\frac{\kappa}{\mu}{\sin( {\mu\; z} )}{A_{2}(0)}{A_{2}(z)}}} = {{{\mathbb{i}}\;\frac{\kappa}{\mu}{\sin( {\mu\; z} )}{A_{1}(0)}} + {( {{\cos( {\mu\; z} )} - {{\mathbb{i}}\;\frac{\sigma}{\mu}{\sin( {\mu\; z} )}}} ){A_{2}(0)}}}}} & (7)\end{matrix}$where μ=√{square root over (σ²+κ²)}. The power of the mode j isdetermined as:P _(j)(z)=∫dxdyE _(j)(x,y,z)E* _(j)(x,y,z)=|A _(j)(z)|²  (8)Here it is assumed that the transverse components of the modes arenormalized:∫dxdye _(j)(x,y)e* _(j)(x,y)=1  (9)It is possible to find the LPG parameters θ, σ, κ, and L, so that, forarbitrary C_(j0) and φ_(j), the requested A_(j)(L) at z=L can beobtained, which satisfy the energy conservation rule:P ₁(L)+P ₂(L)=P ₁(0)+P ₂(0)  (10)whereP _(j)(0)=|A _(j)(0)|² , P _(j)(L)=|A _(j)(L)|²  (11)The corresponding equations for σ, κ, and L are found from Eq. (7):

$\begin{matrix}{{\cos( {\mu\; L} )} = {{Re}\; X}} & (12) \\{{\frac{\kappa}{\mu} = {- \frac{{\mathbb{i}}\; Y}{\sqrt{1 - ( {{Re}\; X} )^{2}}}}}{where}} & (13) \\{X = \frac{{{A_{1}^{*}(0)}{A_{1}(L)}} + {{A_{2}(0)}{A_{2}^{*}(L)}}}{{{A_{1}(0)}}^{2} + {{A_{2}(0)}}^{2}}} & (14) \\{Y = {\frac{{{A_{2}^{*}(0)}{A_{1}(L)}} - {{A_{1}(0)}{A_{2}^{*}(L)}}}{{{A_{1}(0)}}^{2} + {{A_{2}(0)}}^{2}}.}} & (15)\end{matrix}$Eq. (13) is self-consistent only if the right hand side is real. FromEq. (15), the later condition is satisfied ifRe(A* ₂(0)A ₁(L))=Re(A ₁(0)A* ₂(L)).  (16)Eq. (16) can be satisfied with appropriate choice of the LPG phaseshift, θ. Thus, the input modes 1 and 2, with arbitrary amplitudes andphases, can be converted into any other modes, with arbitrary amplitudesand phases, if the condition of the energy conservation, Eq. (10), isfulfilled.

In some applications, it may be necessary to convert two modes withknown input powers, P₁(0) and P₂(0) into two modes with the requestedpower ratio P₂(L)/P₁(L) and with no restrictions on the phases of A₁(L)and A₂(L). This conversion can be performed with the simplified LPG,which satisfies the phase matching condition, σ=0. For example, assumethe condition that after passing the coupling region of length L, thelight is completely transferred to mode 1 and mode 2 is empty:P ₁(L)=P ₁(0)+P ₂(0), P ₂(L)=0, P _(j)(L)=|A _(j)(L)|²  (17)This condition can be satisfied independently of the initial phases ofA₁(0) and A₂(0) only if one of the initial powers is zero. For example,if P₁(0)=0 then Eq. (4) is satisfied ifcos(κL)=0  (18)This result is used in mode conversion based on long period fibergratings. However, if both of initial powers P₁(0) and P₂(0) are notzeros, Eq. (17) can be satisfied when the initial phase differencebetween modes 1 and 2 is

$\begin{matrix}{{\arg( {{A_{1}(0)}/{A_{2}(0)}} )} = {\pm \frac{\pi}{2}}} & (19)\end{matrix}$Then the condition of full conversion of modes 1 and 2 into mode 1 is:

$\begin{matrix}{{\tan( {\kappa\; L} )} = \frac{{\mathbb{i}}\;{A_{2}(0)}}{A_{1}(0)}} & (20)\end{matrix}$The right hand side of this equation is real due to Eq. (19). Thus, inorder to perform essentially full conversion of light, which isarbitrarily distributed between two modes, the initial phases of thesemodes should be adjusted and the coupling coefficient κ and couplinglength L should be chosen from Eq. (20). Furthermore, if the phasecondition of Eq. (19) is satisfied then it can be shown that the powersof modes can be arbitrarily redistributed with the appropriate choice ofcoupling parameters. In fact, assume that the ratio of the input modepowers is R₀=P₁(0)/P₂(0). Then in order to arrive at the output moderatio R_(L)=P₁(L)/P₂(L), the coupling coefficient κ may be defined fromthe equation:

$\begin{matrix}{{{\tan( {\kappa\; L} )} = {\mp \frac{R_{0}^{1/2} + R_{L}^{1/2}}{1 - ( {R_{0}R_{L}} )^{1/2}}}},} & (21)\end{matrix}$where the signs ∓ correspond to ± in Eq. (19). Eq. (20) is derived fromEq. (7) for σ=0. For the condition of full mode conversion, R_(L)=∞, Eq.(21) coincides with Eq. (18). Practically, Eq. (21) can be satisfied bychoosing the appropriate LPG strength and length. Eq. (19) can besatisfied by changing the length of the fiber in front of LPG byheating, straining, or with other type of refractive index perturbationor deformation. Such perturbations and deformations are described inU.S. Pat. No. 6,768,835, which is incorporated herein by reference. Thiscondition can be also satisfied by inscribing the LPG at the properplace along the fiber length.

This basic teaching can be extended to the more general case whereinlight propagating along M modes with amplitudes A₁ ⁰, . . . , A_(M) ⁰ isconverted to the same or other N modes with amplitudes A₁ ^(f), . . . ,A_(M) ^(f). This can be done by a series of two or more mode couplersdescribed above and illustrated in FIG. 4. Due to energy conservation:P ₁ ⁰ + . . . +P _(M) ⁰ =P ₁ ^(f) + . . . +P _(N) ^(f) , P _(j) ^(0,f)−|A _(j) ^(0,f)|².  (22)

Without loss of generality, assume M=N, which can be always done byadding empty modes. If P₁ ⁰ is the largest power among the initialpartial powers and P₁ ^(f) is the smallest power among the final partialpowers then, according to Eq. (22), we have P₁ ⁰≧P₁ ^(f). The firsttwo-mode transformation fills mode 1 with the desired power: P₁ ⁰+P₂⁰→P₁ ^(f)+P₂′ where P₂′=P₁ ⁰+P₂ ⁰−P₁ ^(f). In the result of thistransformation, the problem of conversion is reduced to the case of N−1modes, which can be solved similarly. Thus, with reference to FIG. 4,any power redistribution between two sets of N modes can be performedwith a series 49 of N−1 two-mode transformations as shown in the figure.In the device of FIG. 4 the mode transformers are complex LPGs formed inthe core 48 of optical fiber 47, and arranged serially along the lengthof the fiber.

Alternatively, essentially the same result can be achieved using an LPGwhere the individual gratings 49 in FIG. 4 are superimposed on oneanother. This complex LPG is shown in FIG. 5 where grating 52 issuperimposed on grating 51. Only two gratings are shown for clarity.Also for clarity, grating 52 is shown slightly larger than grating 51.The complex LPG simultaneously performs coupling and transformationsbetween several modes. The LPGs may be chosen to perform couplingbetween mode 1 and all other modes, while the intermode coupling betweenmodes, which have numbers greater than one, is zero. The coupling waveequations, which describe the considered system are:

$\begin{matrix}\begin{matrix}\begin{matrix}{\frac{\mathbb{d}A_{1}}{\mathbb{d}z} = {{\mathbb{i}}( {{\kappa_{12}A_{2}} + {\kappa_{13}A_{13}} + \ldots + {\kappa_{1\; N}A_{N}}} )}} \\{{\frac{\mathbb{d}A_{2}}{\mathbb{d}z} = {{\mathbb{i}}\;\kappa_{12}A_{1\mspace{175mu}}}}\mspace{140mu}}\end{matrix} \\\ldots \\{\frac{\mathbb{d}A_{N}}{\mathbb{d}z} = {{\mathbb{i}\kappa}_{1\; N}A_{1}}}\end{matrix} & (23)\end{matrix}$These equations are the generalization of the coupling mode equations,Eq. (5). The initial power distribution is:P ₁ ⁰ =|A ₁(0)|² , P ₂ ⁰ =|A ₁(0)|² , . . . , P _(N) ⁰ =|A_(N)(0)|²  (24)Solution of Eq. (23) with these boundary conditions leads to thefollowing condition of conversion of all modes into the single mode 1:

$\begin{matrix}{{{\tan( {L\sqrt{\sum\limits_{n = 2}^{N}\kappa_{1\; N}^{2}}} )} = {\frac{\mathbb{i}}{A_{1}(0)}\sqrt{\sum\limits_{n = 2}^{N}\lbrack {A_{n}(0)} \rbrack^{2}}}},} & (25)\end{matrix}$which can be satisfied only under the condition of the phase shifts:

$\begin{matrix}{{{\arg( {{A_{1}(0)}/{A_{n}(0)}} )} = {\pm \frac{\pi}{2}}},{n = 2},3,\ldots\;,N,} & (26)\end{matrix}$Eq. (26) means that the difference between phases of all modes exceptmode 1 should be equal to zero or π, while the difference between thephase of mode 1 and the phases of other modes should be ±π/2. For theparticular case of N=2, Eqs. (25) and (26) coincide with Eq. (20) and(19), respectively. Results show that, using LPG mode transformers, itis possible to convert the arbitrary distributed modes into a singlemode if the phases of modes are appropriately tuned. The phases of LPGscan be tuned by shifting the positions of individual LPGs with respectto each other by, for example, using mechanisms described earlier.

Referring again to FIGS. 4 and 5, the LPGs extend into the cladding asshown. This may be useful if the gratings are to effectively transformhigher order modes propagating outside the core. The output of the modeconverter section is not shown, but is evident in FIGS. 1-3.

It should be understood that the drawing is not to scale. For example,the gain section 47 is typically much longer.

The LPG mode transformers in FIG. 5 may be superimposed completely orpartially. In fabricating devices with superimposed gratings thesuperimposed grating pattern may be formed in discrete steps by formingone grating then superimposing a second, third, etc. gratings on thefirst, second, etc. grating. Alternatively, the superimposed gratingelements may be formed serially in a point to point manner, or may beformed in a single step using a mask pattern comprising superimposedgratings. As stated earlier, for any of these cases where multiple LPGsare used, either arranged serially or superimposed, the LPG may bereferred to as a complex LPG. A complex LPG is defined as a gratinghaving more than one simple LPG, and having more than one distancebetween grating elements. In the serial LPG case the distance will beconstant for the first grating but will change for the next grating. Inthe superimposed grating case, the distance between elements will changemore or less continuously.

The spacing separating the LPGs in FIGS. 4 and 5, and the placement ofthe LPG along the optical fiber are relevant parameters in the operationof the device. These can be tuned in the manner described above. Atuning device is shown schematically at 42. In this case the tuningdevice is shown as a heating element to vary the refractive index of theoptical fiber. Other tuning devices may be used.

The construction and design of LPGs is known in the art. Mode convertersmade using LPGs are described in more detail in U.S. Pat. No. 6,768,835,issued Jul. 27, 2004, which is incorporated herein by reference. Thegratings may be formed by locally changing the refractive index in thecore of the optical fiber to form a pattern of serial regions withaltered refractive index. There are a variety of methods for producingthese changes. A common method is to dope the core of the optical fiberwith a photosensitive agent, such as germanium, and “write” therefractive index pattern using UV light directed on the optical fiberthrough a spatial filter. See for example, U.S. Pat. No. 5,773,486,issued Jun. 30, 1998.

Another effective method is described in U.S. patent application Ser.No. 11/903,759, filed Sep. 25, 2007, and incorporated by referenceherein. This form of LPG is produced by physically deforming the opticalfiber. Embodiments of this type of LPG are represented by FIGS. 6-8.FIG. 6 shows a single LPG with coupling between two modes as indicatedin the associated diagram. The diagram describes energy exchange formodes 1 and 2 as were considered with equation 2 earlier. FIG. 7 showsan embodiment of a complex LPG with multiple gratings 1, 2 to N−1 formedalong the length of the optical fiber.

As in the case of FIG. 5, the individual serially located gratings maybe merged into a single complex grating as shown in FIG. 8.

In the embodiments described above, the HOM section will usually containmultiple modes and the output will typically be a single mode. It shouldbe evident from the description above that any number of input modes canbe processed according to the invention, with one or more output modes.The effect of the mode transformers may be to convert the modes toanother mode, or to increase or decrease the power ratio between themodes.

While the mode transformation illustrated in FIGS. 1-3 is essentially atransformation of multiple modes to a single mode, there may beapplications where the desired output also has multiple modes. Thedesign information that follows is general to all cases of interest.Since the mode converter devices are reciprocal devices, design of a 1to n mode converter is the same as an n to 1 converter, or an m to n orn to m converter.

The mode transformations using LPGs will now be described in detail. Inthis description, the LPG example used is the embodiment of FIG. 5,i.e., superimposed LPGs (SLPGs). It should be understood, as describedearlier, that alternatives to SLPGs may be used.

Consider a coherent beam emerging from the SLPG converter positioned atthe end of a single mode optical fiber, and in particular, an SLPGconsisting of five axially symmetric LPGs that couple six LP_(0j) modes.The output beam generated by a linear combination of these modes isdetermined using the Fresnel diffraction integral in the form of theHankel transform:

$\begin{matrix}{{{E_{out}( {\rho,z} )} = {\frac{2\pi\;{\mathbb{i}}}{\lambda( {z - L} )}{\exp\lbrack {\frac{2\pi\;{\mathbb{i}}}{\lambda}( {z - L} )} \rbrack}{\int_{0}^{R}{{E_{out}( {\rho_{1},L} )}{\exp\lbrack \frac{\pi\;{{\mathbb{i}}( {\rho_{1}^{2} + \rho^{2}} )}}{\lambda( {z - L} )} \rbrack}{J_{0}\lbrack \frac{2{\pi\rho}_{1}\rho}{\lambda( {z - L} )} )}\rho_{1}{\mathbb{d}\rho_{1}}}}}},{\rho = {\sqrt{x^{2} + y^{2}}.}}} & (27)\end{matrix}$Here R is the fiber radius, E_(o)(ρ,L) is the field distribution at theend-wall of the fiber, z=L, which in this case is a linear combinationof six normalized transverse LP_(0j) modes, e_(oj) ^(LP)(ρ):

$\begin{matrix}{{E_{out}( {\rho,L} )} = {\sum\limits_{j = 1}^{6}{A_{j}{e_{0j}^{LP}(\rho)}}}} & (28)\end{matrix}$Modes e_(oj) ^(LP)(ρ) are uniquely determined by the refractive indexprofile of a fiber. In our modeling, we considered an SMF-28 fiber(R=62.5 μm, ρ_(core)=4.1 μm, refractive index difference 0.36%), forwhich these modes were calculated numerically. The coefficients A_(j) inEq. (27) may be optimized to focus the beam in the near field region orto approach a homogeneous beam profile in the far field region.

For the near field case, in order to increase the peak intensity and tosuppress the sidelobes, the beam profile determined by Eq. (27) and (28)can be optimized by variation of the complex-valued coefficients A_(j).In our modeling, the objective function is chosen in the form:

$\begin{matrix}{{F( {A_{2},A_{3},{\ldots\mspace{11mu} A_{6}}} )} = {\int_{\rho_{m}}^{\infty}{{{E_{out}( {\rho,z_{0}} )}}{\mathbb{d}\rho}}}} & (29)\end{matrix}$where we set A₁=1 in the Eq. (28) for E_(out)(ρ,z). Minimization ofF(A₂, A₃, . . . A₆) was performed at fixed z₀−L=0.5 mm and z₀−L=1 mm byvariation of 5 complex parameters A₂, A₃, . . . A₆. The parameter ρ_(m)defines the region outside the central peak of the emerging beam wherethe sidelobes are suppressed. In our modeling, we chose ρ_(m)=15 μm. Theobtained optimum values of A_(j) are given in Table 1.

TABLE 1 Far Focused at 0.5 mm Focused at 1 mm field uniform A₁ 1 1 1 A₂0.60253exp(1.53614i) 1.16235exp(1.61978i) −0.05548 A₃1.07471exp(1.89742i) 1.80893exp(2.33612i) −0.14934 A₄ 1.2775exp(2.47505i) 1.55662exp(−2.82123i) −0.41936 A₅1.11976exp(−2.99339i) 0.64839exp(−1.16064i) −0.93562 A₆0.59115exp(−2.07753i) 0.39395exp(2.13461i) −0.69917

The corresponding filed amplitude profiles at z₀−L−0.5 mm and z₀−L=1 mm(near field) are shown in FIG. 10 as solid and dashed curves,respectively. The lower row of plots demonstrates significantimprovement of the optimized profiles as compared to the profiles ofbeams generated by individual LP_(0j) modes. Note that, for bettervisibility, FIG. 10 shows the filed amplitude rather than the fieldintensity distribution. The relative improvement of the optimized fieldintensity is more apparent.

Table 2 tabulates the fraction of the total beam power inside the 15 mmradius circle at 0.5 mm from the fiber end, and inside the 25 mm radiuscircle at 1 mm from the fiber end, showing the values for the individualmode beams and the optimized beam.

TABLE 2 Foc. at Foc. at LP₀₁ LP₀₂ LP₀₃ LP₀₄ LP₀₅ LP₀₆ 0.5 mm 1 mm Powerinside the 15 μm radius circle at 0.5 mm (%) 21.9 8.2 22.5 30.2 28.632.0 99.0 Power inside the 25 μm radius circle at 1 mm (%) 16.1 32.447.2 42.2 36.4 13.5 98.3

For any of the LP_(0j) modes in Row 2 of Table 2 the power fraction doesnot exceed 32%. However, it approaches 99% for the optimized beam.Similarly, the power fraction for the LP_(0j) modes in Row 3 thisfraction varies between 16% and 43%, but is equal to 98.3% for theoptimized beam. Comparison given by FIG. 5 and Table 2 clearly indicatesthat the suggested SLPG mode converter can serve as an efficient beamfocuser.

For the far field case, which is defined by the inequality z−L>>R²/λ,the integral in Eq. (27) is simplified to

$\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{{{E_{far}( {\theta,r} )} = {\frac{2\pi\;{\mathbb{i}}}{\lambda\; r}\exp\{ {\frac{2\pi\;{\mathbb{i}}}{\lambda}r} \}{f(\theta)}}},}\mspace{85mu}} \\{{{r = \sqrt{( {z - L} )^{2} + \rho^{2}}},}\mspace{31mu}}\end{matrix} \\{{{f(\theta)} = {\int_{0}^{R}{{E_{out}( {\rho_{1},L} )}{J_{0}\lbrack \frac{2\pi\;\theta\;\rho_{1}}{\lambda} \rbrack}\rho_{1}{\mathbb{d}\rho_{1}}}}},}\end{matrix} \\{{\theta = {\frac{\rho}{z - L}{\operatorname{<<}1.}}}\mspace{265mu}}\end{matrix} & (30)\end{matrix}$Here the scattering amplitude f(θ) and the scattering angle θ areintroduced. In numerous applications (e.g. materials processing, laserprinting, micromachining in the electronics industry, opticalprocessing) it is desirable to uniformly illuminate a specific volume ofspace with a laser beam. Following the basic teachings describedearlier, the SLPG converter can be used as simple, robust, and efficientall-fiber beam homogenizer. The sum of LP_(0j) modes generated by SLPG,Eq. (28) forms a beam that has a very uniform central region. To addresshomogenizing of the beam in the far-field region, the objective functionmay be chosen in the form

$\begin{matrix}{{F( {A_{1},A_{2},A_{3},{\ldots\mspace{11mu} A_{6}}} )} = {\int_{0}^{\theta_{m}}{{{{E_{far}( {\theta,r_{0}} )} - E_{0}}}{\mathbb{d}\theta}}}} & (31)\end{matrix}$The function F(A1, A2, A₃, . . . A₆) was minimized by numericalvariation of six real variables A1, A2, A₃, . . . A₆, choosing thehomogenized beam radius, θ_(m), and the field amplitude, E₀ manually.FIG. 9 compares the far-field amplitude distributions for the first sixLP_(0j) modes and their optimized sum. The homogenized beam profile,which is shown in FIG. 9, was obtained for parameters A_(j) given incolumn 4 of Table 1. The central peak of the optimized sum has thediameter of 6.8° and 91% of the total beam power. A homogeneous part ofthis peak, where the relative amplitude nonuniformity does not exceed±0.2%, has the diameter 4° and 52% of the total beam power. Thus, theconsidered example demonstrates that SLPG consisting of a reasonablenumber of gratings can produce extremely homogeneous light beams.

Coefficients A_(j) in Eq. (28) determine the superposition of outputfiber modes to form a beam of the desired shape. The specific design ofSLPGs using the example of the homogenized beam considered above may beimplemented by determining the SLPG refractive index variation given by:

$\begin{matrix}{{\delta\;{n( {x,y,z} )}} = {\lbrack {{\delta\; n_{0}} + {\sum\limits_{k = 2}^{6}{\delta\; n_{1\; k}\;{\cos( {{2\;\pi\;{z/\Lambda_{1\; k}}} + \phi_{1\; k}} )}}}} \rbrack{\theta( {\rho_{core} - \sqrt{x^{2} + y^{2}}} )}}} & (32)\end{matrix}$

with parameters Λ_(1k), φ_(1k), δn_(1k), and δn₀. Coefficients A_(j) forthis example are given in column 4 of Table 1. Other parameters thatdetermine the SLPG are summarized in Table 3.

TABLE 3 j 1 2 3 4 5 6 β_(j) μm⁻¹ 5.87187 5.86328 5.86254 5.8613 5.859585.85739 I_(1j) — 0.03226 0.05863 0.07999 0.09643 0.10849 I_(jj) 0.745110.00145 0.00481 0.00907 0.01347 0.01762 10⁵κ_(1j) μm⁻¹ — 0.07952 0.214010.60098 1.34084 1.00199 10⁵ δn_(1j) — 1.21617 1.80099 3.70689 6.86044.55656 10⁵κ_(jj) μm⁻¹ 27.3969  0.05319 0.17686 0.3334 0.4954 0.64779Λ_(1j) — 731.132 673.327 594.553 511.214 433.809 φ_(1j) — 0.4653120.527147 0.605417 0.686417 0.762612The values in the Table 3 are calculated as follows. First, thepropagation constants of the LP_(0j) modes of an SMF-28 at wavelengthλ=1.55 μm are given in row 2. The overlap integrals I_(1j) and I_(jj)for these modes are calculated based on the wave equations using thefollowing.

Assume the SLPG is introduced in the core of the optical fiber by aperturbation of the refractive index,

$\begin{matrix}{{\delta\;{n( {x,y,z} )}} = \lbrack {{ \quad{{\delta\; n_{0}} + {\sum\limits_{{j > k} = 1}^{N}{\delta\; n_{jk}{\cos( {{2\;\pi\;{z/\Lambda_{jk}}} + \phi_{jk}} )}}}} \rbrack{\theta( {\rho_{core} - \sqrt{x^{2} + y^{2}}} )}},} } & (33)\end{matrix}$where x and y are the transverse coordinates, z is the longitudinalcoordinate, θ(s) is a Heaviside step function, ρ_(core) is the coreradius, and Λ_(jk) are the periods of harmonics. In the coupled wavetheory of a weakly guiding fiber, the field can be written in the scalarform E(x,y,z)=Σ_(j)A_(j)(z)exp(iβ_(j)z)e_(j)(x,y), where e_(j)(x,y) arethe transverse components of eigenmodes and β_(j) are the propagationconstants. It is assumed that the periods Λ_(jk) approximately match thedifferences between the propagation constants of the fiber modes, i.e.,2π/Λ_(jk)≈β_(j)−β_(k), so that the harmonic (j,k) couples together modesj and k. The coupled mode equation for A_(j)(z) can be derived from ageneral coupled mode theory in the form:

$\begin{matrix}{{\frac{\mathbb{d}A_{j}}{\mathbb{d}z} = {{\mathbb{i}}{\sum\limits_{k = 1}^{N}{\kappa_{jk}{\exp\lbrack {{{{\mathbb{i}}( {\beta_{j} - \beta_{k} + \frac{2\pi}{\Lambda_{jk}}} )}z} + {{\mathbb{i}}\;\phi_{jk}}} \rbrack}A_{j}}}}},} & (34)\end{matrix}$where κ_(jk) are the coupling coefficients defined by the followingequations:

$\begin{matrix}\begin{matrix}\begin{matrix}{{\kappa_{jj} = {\frac{\pi\;\delta\; n_{0}}{\lambda}I_{jj}}},} \\{{\kappa_{jk} = {\frac{\pi\;\delta\; n_{jk}}{\lambda}I_{jk}}},{j \neq k},}\end{matrix} \\{I_{jk} = {\int\limits_{\sqrt{x^{2} + y^{2}} < \rho_{core}}{{\mathbb{d}x}{\mathbb{d}y}\;{e_{j}( {x,y} )}{{e_{k}( {x,y} )}.}}}}\end{matrix} & (35)\end{matrix}$Here, λ is the wavelength of light in free space and the transverseeigenmodes e_(j)(x,y) are normalized,

∫_(−∞)^(∞)∫_(−∞)^(∞)𝕕x𝕕y e_(j)²(x, y) = 1.In Eq. (34), it is assumed that the periods Λ_(jk) are positive for j>kand the propagation constants are monotonically decreasing with theirnumber, i.e. β_(j)<β_(k) for j>k. In addition, φ_(jj)=0, 1/Λ_(jj)=0,Λ_(jk)=−Λ_(kj), φ_(jk)=−φ_(kj), and, as follows from Eq. (35),κ_(jk)=κ_(kj).

The values of these integrals are given in rows 3 and 4.

Next, the following equation gives values for the phase shifts, φ_(ij),the relative values of coupling coefficients, κ_(1j)/κ₁₂, and thecorresponding SLPG length L.

$\begin{matrix}{{\phi_{1\; j} = {\gamma_{1} - \gamma_{j} - {\kappa_{11}L} + {{\kappa_{jj}L} \pm \frac{\pi}{2}}}},{\kappa_{1\; j} = {\kappa_{12}\frac{{A_{j}(L)}}{{A_{2}(L)}}}},{{\tan( {\mu\; L} )} = {\pm {\frac{{{A_{2}(L)}}\mu}{{{A_{1}(L)}}\kappa_{12}}.}}}} & (36)\end{matrix}$From this equation, the coupling coefficients κ_(1j) should beproportional to A_(j), i.e. κ_(1j)=CA_(j), with a constant C to bedetermined. In theory, Eqs. (36) allow the length of SLPG, L, to bechosen independently of other parameters. However, smaller L requiresstronger gratings and includes less LPG periods. Assuming a reasonablevalue of L=50 mm. Then with A_(j) from Table 1, column 4, we findC=1.4331×10⁻⁵ μm⁻¹ and the values of coupling coefficients κ_(1j)=CA_(j)(j>1) given in row 5 of Table 3. With the known κ_(1j) and I_(1j), fromEq. (35), we find δn_(1j)=λκ_(1j)/(πI_(1j)) at λ=1.55 μm, which is givenin row 6. Index δn₀ may be determined from the condition that theoverall introduced index variation should be positive:δn₀=Σ_(j>1)|δn_(1j)|=1.8141×10⁻⁴. This value of δn₀ together with I_(jj)from row 4 determines the self-coupling coefficients given in row 7.

The periods of gratings, Λ_(1k), in row 8 may be determined from thefollowing:2π/Λ_(jk)=β_(k)−β_(j)−κ_(kk)+κ_(jj),  (36)where the self-coupling coefficients, κ_(jj), and propagation constants,β_(j), are given in row 6 and 1, respectively. Finally, the LPG phaseshifts given in row 9 may be calculated from:φ_(1j)=(κ_(jj)−κ_(ll))L+π/2  (37)

It should be evident to those skilled in the art that a first E-fieldcomprised of a set of modes propagating in an HOM fiber can be matchedto a second, different E-field in second, different medium at the outputof the fiber. By the principle of reciprocity, the light propagation isthe same out of, or into, the HOM fiber. Thus, if, for example, aGaussian beam in the second medium impinging onto an HOM fiber breaks upinto a set of modes of the fiber then a set of LPGs or indexperturbations which generates precisely this set of mode amplitudes andphases in the HOM fiber will cause the light propagating in the otherdirection to recombine at the output to produce a second E-field in theform of the same Gaussian beam in the second medium after the fiberoutput. Moreover, if one can do approximately this linear combination ofmodes, then one can generate at the HOM fiber output a second E-fieldpattern in the medium after the output that is largely the same as thedesired E-field.

This proves that the HOM can go to N modes after the LPG and that theseN modes can give a Gaussian, LMA, or ACPGB mode.

The specific waveguides in the embodiments shown in the figures areoptical fiber waveguides. However, the equations given above are generalwaveguide equations and apply to other forms of waveguides as well. Forexample, the invention may be implemented with planar optical waveguidesin optical integrated circuits. These options may be described using thegeneric expression optical or electromagnetic field waveguide.

In devices made according to the invention the transmission medium, forexample, an LMA optical fiber or free space, coupled to the complex LPGmode transformer will be adapted to support a light beam with an E-fieldsubstantially different (typically larger) than the E-field of the HOMin the HOM optical fiber. Substantially different in this context meansdifferent by at least 25% in area.

Likewise when the properties of the HOM optical fiber and the large modearea optical fiber are expressed in terms of the relative size of theoptical fiber cores, substantially larger in that context means largerby at least 25% in area.

Various additional modifications of this invention will occur to thoseskilled in the art. All deviations from the specific teachings of thisspecification that basically rely on the principles and theirequivalents through which the art has been advanced are properlyconsidered within the scope of the invention as described and claimed.

1. A method comprising: a) transmitting a light beam with at least onehigher order mode (HOM) in a core of a first optical waveguide, wherethe light beam in the first optical waveguide has a first E-field, b)using a complex long period grating (LPG) mode transformer to transformat least a portion of the light beam into one or more modes in the firstoptical waveguide to produce an output light beam, and c) coupling theoutput light beam to a large mode area (LMA) waveguide with a secondE-field, the LMA waveguide having a core that is substantially largerthan the core of the first optical waveguide, wherein a superposition ofthe one or more modes of the output light beam essentially matches afundamental mode of the LMA waveguide.
 2. The method of claim 1 whereinthe first optical waveguide is an optical fiber and the LMA waveguide isan LMA fiber.
 3. The method of claim 1 wherein the complex LPG modetransformer comprises at least 3 LPGs.
 4. An optical device comprising:a) a first optical waveguide having a core and a cladding, with the coreadapted to support a light beam with at least one higher order mode(HOM), the HOM having a first E-field, b) a complex long period grating(LPG) mode transformer coupled to the first optical waveguide andadapted to transform at least a portion of the HOM to one or more modesin the first optical waveguide to produce an output light beam, and c) alarge mode area (LMA) waveguide coupled to the complex LPG modetransformer, the LMA waveguide having a core that is substantiallylarger than the core of the first optical waveguide, the LMA waveguideadapted to support a light beam with a second E-field, wherein thesuperposition of the one or more modes of the output light beamessentially matches the fundamental mode of the LMA waveguide.
 5. Theoptical device of claim 4 wherein the first optical waveguide comprisesan optical fiber and the LMA waveguide comprises an LMA fiber.
 6. Theoptical device of claim 4 wherein the complex LPG mode transformercomprises at least 3 LPGs.
 7. The optical device of claim 6 wherein thecomplex LPG mode transformer comprises photoinduced LPGs.
 8. The opticaldevice of claim 6 wherein the complex LPG mode transformer comprisesLPGs comprising physical deformations of an optical fiber.
 9. Theoptical device of claim 7 wherein the complex LPG mode transformercomprises superimposed LPGs.